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Location: http://www.cs.mun.ca/~ulf/gloss/meta.html. By Ulf Schünemann since 2001. Please mail any comments.

Meta-Semiotics, Reasoning, and Identities

  1. L2: Meta-Signs and Modes of Supposition
  2. Can Second-Order Quantification Be Defined?
  3. Mixed Mode/Level Reasoning
  4. Referential Transparency

L2: Meta-Signs and Modes of Supposition

Meta-Signs (L2)

Signs can mean other signs, ie. be secondary signs or meta-signs. For instance, the Morse code ·- "means" the letter A. The meaning of secondary signs is of a different quality than the meaning of word in the language. It is a meta-semiotic meaning, the reference of one (secondary) sign to a (primary) sign. They are thus meta-signs, without being directly signs. [HSem 200].

Meta-Language (L2)

If all things are considered at semantic level 0, then statements about these things, like "it snows", "Donald Duck visits Mickey Mouse", are at the object level (semantic level 1), and the language to talk of such statements is the object language [DvR]. Statements talking about (object-level) statements, e.g., about their truth, are at semantic level 2: meta-level statements. The language to talk about things and object language statements is the meta-language. E.g., «[i]n the sentence, 'In Early Middle English, the word eyren meant "eggs"', the object language (the language under discussion) is Early Middle English, and the metalanguage is Modern English» [x]. E.g., "it snows" is a true statement, or "Donald Duck" is the name of a character by Walt Disney. «The fact that words are sometimes used in this manner is noted by the medieval William of Sherwood, and Leibniz repeatedly mentiones it. Frege, in his discussion of sense and reference, gives the following formulation:

Modes of Supposition

Statements may be made of a subject term on different levels (of form, content, and denotation). «Mediæval logicians referred to this facet of statements as modes of supposition, or what is 'supposed' by the subject in relation to each predicate» [AUOOP]. Supposition theory was developed in the late twelfth century [x], preserved in handbooks by Peter of Spain, Lambert of Auxerre, and William of Sherwood [x],
  1. Refering to a term's denotation (object level):
    Personal Supposition: «A term has personal supposition when it is used to talk about what it signifies, for example 'lion' in 'She is feeding the lion now'» [x].
  2. Refering to a term's sense (meta-level):
    Logical or Formal Supposition: «A term has formal supposition when it stands for a concept or a universal, as in 'Lion is a species'» [x]. «[T]he subject ... stands for the nature of the subject alone. It does not stand for any of the individuals of the species but only for the concept; it does not refer to real beings (i.e. for the intension only). ... 'Man is a species of animal', for example, cannot be said of individual men because what is supposed by 'man' in this case, is only the logically universal nature.» [AUOOP].
  3. Refering to a term's denotation and sense(?):
    Real Supposition: «[T]he subject in relation to what is predicated of it, stands for the nature of the species and for all of the members of the species (i.e., for the intension and the extension). If we say that 'man is rational', the term 'man' in this context, stands both for human nature and for all of those ... individuals in which that nature is found» [AUOOP].
  4. Refering to a term's form (meta-level):
    Material Supposition: «A term has material supposition when it is used autonomously to talk about its inscription or utterance, as in 'Lion has four letters'» [x]. «[T]he subject ... stands only for the compositon of the word itself rather than for the logical or the real nature which the word can signify» [AUOOP]. E.g. or 'Man' is a three-lettered word. Material supposition may be indicated in written language by the use of quotes.

  5. Refering to the term as a linguistic element (meta-level):
    Lexical Supposition I stipulate there is another meta-level supposition mode by which one refers, neither to the form, sense, nor denotation, but to the word as a symbol with an identity, traceable through history, to the word as an element of the lexicon of a language, i.e., to the lexical word or lexeme.
    ->  lexical word

    No other supposition mode seems to match for the occurence of 'table' in Table is a Latin loan word. The same for 'Potenzial' in Potenzial was spelled Potential in German before the spelling reform.
Cf. modes of supposition in ->  programming languages
Cf. Carnap's formal and material mode [x].

Mode-classified Definitions

A particular kind of statement is the explanation or definition "x is X" of a word x. («The Word «Definition» refers to nearly every answer to the question «what is x?» ...» [ZDM].)
  1. Material definitions: x in material supposition, X describes x's form: E.g. Taf is a three-letter word.
  2. Meaning analysis: x in formal supposition, X describes its sense, no denotation is needed. The meaning of a term (`analysandum') already existing in the language is analyzed to determine its ``logically'' necessary and sufficient properties (`analysans') [DvR]. A unicorn is a horse-like animal with a single straight horn on its head is a meaning analysis which is an analytically true statement IMHO.
  3. Empirical analysis: x in personal/real supposition, X describes its denotation. In the world the denotation of a term (`analysandum') is analyzed to determine the ``factual'' properties (`analysans') [DvR]. The empirical analysis [the denotation of] `life form' (analysandum) is a `DNA based reproductive system' (analysans) is an a-posteriori true statement IMHO (according to what we know of the world up to now). Finding a correct (commonly aggreed upon) meaning analysis for the term "life form" may be much more difficult.

Can Second-Order Quantification Be Defined?

The classical view of second-order quantification in the induction principle has to be adopted to exclude non-standard models. «But then it is impossible to axiomatize the logic; no effective formalization can capture all the truths. If we go back and adopt the Henkin interpretation then the logic has a complete formalization.
So there are two different stories that could be told about a second-order formalization of arithmetic. On one account, the second-order quantifiers range over uncountable sets of predicates and the model of arithmetic is the standard one, but any formalization of it will be incomplete and therefore fail to completely capture this meaning. This makes sense if we take the meaning of a formalism to be determined by its human users. The human can simply declare that her quantifiers are to be interpreted classically, that her arithmetic is standard, and that the lack of completeness of the axiomatic framework is of minor concern to her. From the other perspective, the formalism's meaning can be determined only by its axiomatic framework, and the second-order quantifiers range over the properties which the formalism can describe. On this view, it is simply true that any formal arithmetic has nonstandard models, and no amount of bluster by a human user can make it otherwise» [APKR].

Cf. source of symbol meaning.

Mixed Mode/Level Reasoning

«Supposition becomes a serious concern when statements are used in an argument, because terms in an argument must not only keep the same signification [meaning?], but also the same supposition» [AUOOP]. Switching modes of supposition may produce fallacies:
Man is a species of animal. 'Man' is a three-lettered word.
Socrates is a man. Socrates is a man.
Therefore, Socrates is a species of animal.  Therefore, Socrates is a three-lettered word.

Alfred Tarski 1935 investigated meta-level statements of the following general scheme [DvR]:

reference-to-p is a true statement iff p
The most obvious form of reference to a statement p is the quoted p, yielding
"p" is a true statement iff p
This looks like a logical truth. But be cautious:
Natural language is "semantically closed" -- it is its own meta language. Especially, it allows to construct statements (sets, formulae) which (indirectly) refer to themselves. This gives raise to some (famous) paradoxes: The dubious statements in all these examples mix semantic levels. In the scientific discourse, in order to avoid such antimonies, sentices mixing semantic levels should be rejected as `semantic nonsense' [ZDM]. One should separate the object language from the meta language. In the meta-language quotation marks are used around an object-language term t to refer not to what t refers to, but to refer to term t itself. Then cat is an animal and "cat" consists of three letters are easily recognized as correct, while "cat" is an animal and cat consists of three letters are obviously wrong, since "cat" is not an animal, but a word, and since cat is not a word, but an animal and thus does not consist of letters [ZDM]. The object language is made free of constructs for refering to, and talking about the meaning of, its own statements (esp., it does not contain quation marks). And to talk about the meta language one needs a meta meta language, and so on. Russel's theory of types is the application of this leveling to set theory.

Referential Transparency

From [RTDU]: «The notations of referential transparency and referential opacity are common in discussions of properties of programming languages. They were originally suggested by Quine and brought into computer science by Landin and Strachey. The notation however have changed during time and the formal or informal definitions found in the literature are not equivalent.»

1. Examples of referential transparency: From [Stansifer]: «The phrase "referential transparency" was first used by Whitehead and Russel in Principia Mathematica to compare the following two syllogisms:
All men are mortal; Everything Xenophon said about Socrates is true;
Socreates is a man; Xenophon said: "Socrates is mortal";
Therefore Socrates is mortal.  So Socrates is mortal.
...

Willard Quine uses the phrase slightly differently to refer to the substitutivity of identities [terms with the same denotation]. For example, in the sentence the word "Tully" may be replaced by "Cicero," which was another name of the same man.»

2. Referential opacity based on material supposition: In the phrases

  1. "Cicero" contains six letters [RTDU]
  2. Peter the Great was so-called because of his exploits [Stansifer]
    = Peter the Great was called "Peter the Great" because of his exploits
  3. Philip believes [or says] that Tegucigalpa lies in Nicaragua [RTDU]
    = Philip believes [says] "Tegucigalpa lies in Nicaragua" (quoted direct speech instead of indirect speech, this is a transformation with unchanged deep structure)
the statements' truth (denotation) is changed by replacing the terms "Cicero", "Peter the Great" and "Tegucigalpa" by other terms like "Tully", "Ivan Ivanovitch" and "the capital of Honduras" which de-facto denote the same respective objects (an a-posteriori truth). This could happen because of the statements refer to the (form or content of) terms, i.e., are meta-level statements. «In this way the quotes change or destroy reference, that is, the relation between a term and the object(s) it denotes.» [RTDU].
It is possible to change the transformed (2) to Ivan Ivanovitch was called "Peter the Great" because of his exploits. And it might also be acceptable to change the original (3) to Philip believes that the city which - as educated people know - is Honduras's capital, lies in Nicaragua.
In programming languages, referentially opaque constructs, which depend on their argument's form, not value, are quotes for character strings, quote in Lisp [RTDU], and C macros like assert which apply the ``stringification'' operator '#' to their argument expressions.

3. Other cases of referential opacity: Intension, propositional attitude, etc., may destroy referential transparency (cf. extensionality below).

  • Note that (the typical examples of) believes fall under material supposition (see case (3) above).
    1. Fictious entities: In I want to visit the lost city of Atlantis "Atlantis" refers to a fictious place (it has a sense, but no denotation, see above).
    2. Necessity: Aristotle necessarily was Aristotle is a logical truth, whereas Aristotle necessarily was the teacher of Alexander is false (it is an empirical fact that Aristotle was the teacher of Alexander, but that is not a necessity).
    3. Causality: He coughed because he smoked is intensional [x] - replacing "he smoked" by other true sentences is not truth preserving. (cf. it would work for material implication "he smoked -> he coughed")
    4. «From the fact that P thinks of someone as the author of Waverley, it does not follow that he thinks of someone as the author of Ivanhoe, even though the two descriptions are descriptions of one and the same persion, namely, Scott. ... What it shows is that the description the author of Waverley is not the same entity as the description the author of Ivanhoe, even though the person described is the same in both cases. ... To emphasize that it is not the descriptions but the entities described which are said to be identical, one should paraphrase ... `The author of Waverley is the same as the author of Ivanhoe,' [as] some such thing as `One and the same person is the author of Waverley and the author of Ivanhoe'.» [OntRed 50f]
    5. Because conceptual possibility of formulae is a relational property, «a statement of the form "It is possible that p" is not at the same level as p itself but is a metastatement» -> more
    «Possible explanations [for referentially opaque contexts] are: the expression does not really refer (Russell) [cf. fictious entities], or refers to something else (Frege, perhaps Aristotle), or does more than refer (Quine).» [x]

    4. Leibniz's law: «There is ... the general criterion of identity which holds for all entities whatsoever, namely, Leibniz's principle, which states that two entities are the same if and only if they share all their properties (including relations). This law holds for states of affairs as well as for individual things, properties of individual things, classes, etc.» [OntRed 40]

    «Leibniz's law holds for intentional properties and modal properties as well as for "normal" ones. But this fact is easily overlooked if one does not realize that what a definite description expression represents is a description and not the entity described. Consider the description the author of Waverley. There is no question what it is that is described as the author of Waverley; it is obviously a certain person, namely, Scott. But what does the description expression represent? What is it that is the same when we do not have two different descriptions of Scott but, rather, the same description in different linguistic disguises? Frege was the first to raise this question and his answer is by now famous. He holds that description expressions express a sense in addition to referring to a referent.» [OntRed 51]
    «Our considerations show that if one distinguishes properly between descriptions [intensions] and what they describe [denotations], it becomes quite obvious that there are no exceptions to Leibniz's law. There is no property (or a relation), no matter how esoteric or abstract, that belongs to e1 but not to e2 if e1 is the same as e2.» [OntRed 53]

    5. The notion of substitutivity: Continuing with [Stansifer]: «We see the same idea reflected in Frege's Über Sinn und Bedeutung:

      The meaning [denotation] of a sentence must remain unchanged when a part of the sentence is replaced by an expression having the same meaning [denotation].»
    This "substitutivity of identities" (cf salva veritate) is also called Leibniz's law in [RTDU]. Languages where substitutivity fails are characterized as intensional [x].
    Term substitution vs text substitution [my own remark]
    The notion of substitution used for referential transparency is term substitution, not text substitution: Subtituting "7 + 5" for "12" in "1 × 12 × 2" changes the expression's value from 24 to 17 ("1 × 7 + 5 × 2"). Nevertheless, this does not prove that "1 × _ × 2" is a referentially opaque context, since the above substitition changed the term structure. If we substitute "(7 + 5)" for "12" then the term structure does not change, and the value is still 24 ("1 × (7 + 5) × 2").
    Substituting 12 in calc(12) to yield calc(7 + 5) is proper term substitution. However, if calc is the definiens introduced by a nominal definition whose meaning is defined through textual substititution of arguments into the definiens (as it is the case for macros of the programming language C), then calc can be a referentially opaque context for its arguments: 
    #define calc(x) (1 * x * 2)
calc(12)    // value 24
calc(7 + 5) // value 17

    6. Extensionality/Intensionality: There are some notions similar and/or related to referential transparency, often not clearly distinguished from referential transparency [RTDU]:

    Extensionality of an expression is the property that «if we wish to find the value [denotation] of an expression which contains a sub-expression, the only thing we need to know about the sub-expression is its value [denotation]» [Strachey quoted from RTDU].
    «A context or form of words is intensional if its truth is dependent on the meanings, and not just the reference, of its component words, or on the meanings, and not just the truth-value, of any of its subclauses».
  • Note that modality does not necessarily mean to give up extensionality. «There are some interpretations of modal systems where the only departures from the standard semantics is that predicates are assigned different extensions in different worlds. There is still a reduction of properties to extensions» [x].
    Stansifer does not sharply distinguish referential transparency and extensionality when continuing «A language in which the context does not affect the meaning of expressions is said to be referentially transparent. Of course, this is not a precise statement, as context and meaning are both ill-defined» [Stansifer].

    7. Definiteness (context-freedom): «We tend to assume automatically that the symbol x in an expression such as 3 x2 + 2 x + 17 stands for the same thing (or has the same value) on each occasion it occurs» [Strachey quoted from RTDU]. Here only those occurences of symbol x are meant which are in the scope of the same definition of x. (This qualification is necessary for languages allowing to introduce the same symbol name several times but only for certain (non-overlapping) parts of the text).

    Stansifer [Stansifer]: does not sharply distinguish referential transparency and definiteness when continuing «The notion of referential transparency is significant, as it captures the property taken for granted in mathematical language, that expressions denote the same values regardless of context. This property is violated in imperative programming languages because of the central importance of locations shifting values over time.»

    8. Unfoldability of the instantiation of generics (the application of lambda terms): Let f be a generic (a function) with parameter x defined by f(x) = F[x] where F is an expression in which x may occur. Unfoldability means that for all expressions E, the denotation of f(E) is the same as the denotation of F[E] where E has been substituted for x in F.

    Ulf Schünemann 080402